Copy/paste in edit mode in Blender 2.8
I suggest that instead of trying to copy and paste the bevel as a new object, you should simply duplicate the mesh you have, leaving one copy the same, and using the other mesh to create the bevels you wanted.  Simply create the bevels, and with the bevels still selected, hit ctrl-i to invert the selection, and press delete.  This way you end up with just the beveled mesh, yet you still have the original mesh to keep.

Hope that helps.